6 INTEGRATION ALONG A ROAD 217

a number M such that, for every frontier point x $ E of A, and every e > 0, there is
a neighborhood V of x in Cp such that |/(z)| ^ M + e for z e A n V. Show that
|/(z)| < M for every z e A. (One can suppose that \g(z)\ < 1 for z e A. Consider the
function |/(z)| * |#(z)|a, where a > 0 is arbitrary, and apply the result of Problem 9(b)
to that function.)

(b) Show that the result of (a) does not hold if the assumption that /is bounded in A
is deleted (consider the function exp(exp((l — z)/z)) and use Problem 3(b)).
15. Let co(x) be a real function denned in [0, 4- oo [, such that

a>(x) > 0 and lim o>(;c) = + ox

JC-+

Show that if a complex-valued function / is analytic in a neighborhood of the
closed half-plane A: ^(z)^0, then there is at least one point £eA such that
!/(£)!< exp(co(|£|)|£|). (Use contradiction: if the conclusion was not true, prove
that the function \ez\ • |/(z)|~8 would be <1 in A, for every value of e>0, by
applying Problem 9(a).)

16. Let A be an open relatively compact subset of Cp, / a complex- valued function,
analytic in A. Suppose there exists a number M > 0 and a complex- valued function g,
analytic in A, such that g(z) ^ 0 for any z e A, and having the following property:
for every point x of the frontier of A, and every e > 0, there is a neighborhood V of jc
such that |/(z)| < M|#(z)|e for ze A n V. Show that |/(z)| ^ M in A ("Phragmen-
Lindelof 's principle"; use Problem 9(b)).

17. Let U be the open set defined in Problem 3(b), and suppose /is a complex valued
analytic function in a neighborhood A of U, having the following properties:
(1) |/(z)| ^ 1 on the frontier of U; (2) there exists a constant a such that 0 < a < 1
and |/(z)| ^ exp(exp(a^(z))) for z e U. Prove that |/(z)| ^ 1 in U. (Remark that

1

2+1

transforms U into a relatively compact set, and use Phragmen-Lmdelof 's principle
(Problem 16) with g(z) of the form exp(exp(£z)).)

6. INTEGRATION ALONG A ROAD

A path in C is a continuous mapping y of a compact interval I = [a, b] ^ R,
not reduced to a point, into C; if y(I) c A c: C, we say that y is a path in A;
y(a) (resp, y(b)) is called the origin (resp. the extremity) of the path, both
points are also called the extremities of y; if y(d) = y(b), y is called a loop;
if y is constant in I, we also say that the path y is reduced to a point. The
mapping y° of I into C such that y(t) = y(a+b—t) is a path which is
said to be opposite to y. Let Ix = [6, c] be a compact interval in R whose
origin is the extremity of I, and let I2 = I u It — [a, c]; if yl is a path defined
in Ils and such that y^b) = y(b)9 and if we define y2 to be equal to y in I,
to V! in I1? y2 is a path which we denote y v yl9 and which we call the juxta-
position of y and yt.b, c, d are chosen such that G(B) is the half-plane defined by